
For any \(a\in \mathcal {A}\) , the Lagrangian dual problem of the minimization problem in (REF ) is (see [1]}):
\(\begin{aligned}\max _{\lambda _L, \lambda _U, \nu } \quad &g(a,\lambda _L,\lambda _U, \nu )\\\mathrm {s.t.:} \quad &\lambda _L \succeq 0, \text{ }\lambda _U \succeq 0, \text{ }\nu \in \mathbb {R}\end{aligned}\) 
